Why Linear Dependence Matters
Linear dependence tells whether vectors carry repeated direction information. A dependent set has at least one vector made from the others. An independent set adds a new direction. This calculator turns that idea into rank, pivots, and a relation. It helps with bases, spans, subspaces, and matrix problems.
What the Calculator Checks
The tool builds a matrix from your vectors. It can read vectors as rows or as columns. Row reduction then finds pivot columns. The rank is the number of pivots. The vector count is compared with that rank. If rank is smaller than the vector count, the set is dependent. If both numbers match, the set is independent.
Advanced Inputs
You can enter decimals, negative values, fractions written as decimals, and scientific notation. The tolerance field helps with rounded data. A smaller tolerance is stricter. A larger tolerance treats tiny values as zero. This is useful when values come from measurements, simulations, or copied software output.
Reading the Result
The summary shows the dimension, vector count, rank, pivot columns, and free variables. A free variable means one vector can be combined with others. When dependence is found, the calculator gives a nonzero coefficient relation. A relation like c1v1 plus c2v2 equals zero proves dependence.
Using the RREF Table
The reduced row echelon form shows the row operations result. Pivot columns point to essential vectors. Nonpivot columns point to dependent choices. In a basis problem, choose the original vectors that match pivot columns. Do not choose the reduced rows as your basis vectors.
Exports and Study Use
The CSV export is useful for spreadsheets. The PDF export is useful for notes, homework checks, and reports. Both downloads use the displayed result. You can also copy the example data, edit values, and test another set.
Common Mistakes
Do not compare vectors only by length. Direction and combination matter. A longer vector can still depend on shorter vectors after scaling and addition.
Best Practice
Keep each vector the same length. Use one row for each vector when row mode is selected. Use one matrix row per component when column mode is selected. Check rounding before making a final conclusion. For exact algebra, enter clean integers when possible.